A010878 -id:A010878 - cp's OEIS frontend

A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 66, 67, 69, 72, 76, 81, 87, 94, 102, 111, 121, 132, 132, 133, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 198, 199, 201, 204, 208, 213, 219, 226, 234, 243, 253, 264, 264, 265, 267, 270, 274, 279, 285, 292, 300

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 12, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

List([0..60], n-> Sum([0..n], k-> k mod 12 )); # G. C. Greubel, Sep 01 2019

[&+[(k mod 12): k in [0..n]]: n in [0..60]]; // G. C. Greubel, Sep 01 2019

seq(coeff(series(x*(1-12*x^11+11*x^12)/((1-x^12)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 01 2019

Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,3,6,10,15,21,28,36,45,55,66,66},60] (* Harvey P. Dale, Jan 16 2024 *)

a(n) = sum(k=0, n, k % 12); \\ Michel Marcus, Apr 29 2018

[sum(k%12 for k in (0..n)) for n in (0..60)] # G. C. Greubel, Sep 01 2019

a(n) = 66*floor(n/12) + A010881(n)*(A010881(n) + 1)/2.
G.f.: (Sum_{k=1..11} k*x^k)/((1-x^12)*(1-x)).
G.f.: x*(1 - 12*x^11 + 11*x^12)/((1-x^12)*(1-x)^3).

A141726 Sawtooth with period length 9: repeat 8, 7, 6, 5, 4, 3, 2, 1, 0.

Original entry on oeis.org

8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3

Offset: 1

Paul Curtz, Sep 13 2008

Continued fraction expansion of (23342+5*sqrt(44403565))/6961.

Decimal expansion of 973936900/111111111.

Cf. A010878.

PadRight[{},120,Range[8,0,-1]] (* Harvey P. Dale, Dec 23 2020 *)

def A141726(n): return (0, 8, 7, 6, 5, 4, 3, 2, 1)[n%9] # Chai Wah Wu, Jan 10 2023

a(n) = a(n-9).

G.f.: -x*(8+7*x+6*x^2+5*x^3+4*x^4+3*x^5+2*x^6+x^7)/((x-1)*(1+x+x^2)*(x^6+x^3+1)).

Index in c-sequence corrected by R. J. Mathar, Sep 11 2009

A007960 Positive numbers k with the property that some permutation of the digits of k is a triangular number.

Original entry on oeis.org

1, 3, 6, 10, 12, 15, 19, 21, 28, 30, 36, 45, 51, 54, 55, 60, 63, 66, 78, 82, 87, 91, 100, 102, 105, 109, 117, 120, 123, 132, 135, 136, 147, 150, 153, 156, 163, 165, 168, 171, 174, 186, 190, 201, 208, 210, 213, 231, 235, 253, 258, 267, 276, 280, 285, 300, 306, 307

Offset: 1

R. Muller

Leading zeros may be omitted from the permutation of the digits of k to get T. But the number of digits of T must be <= the number of digits of k. - N. J. A. Sloane, Dec 14 2007

Working modulo 9, A010878(A000217(j)) is in the set {0, 1, 3, 6} for all j, never in {2, 4, 5, 7, 8}. Since permutation of decimal digits does not change values mod 9, A010878(n) is also one of {0,1,3,6}. - R. J. Mathar, Jan 08 2008

			Contains k=1, k=10, k=100, etc. derived from T=1.
Contains k=3, k=30, k=300, etc. derived from T=3.
Contains k=15, k=51, k=105, k=150, etc. derived from T=15.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions!

Cf. A000217.

q:= n-> ormap(issqr, map(x-> 1+8*parse(cat(x[])),
        combinat[permute](convert(n, base, 10)))):
select(q, [$1..500])[];  # Alois P. Heinz, Aug 22 2021

Select[Range[500], Length[Select[FromDigits/@Permutations[ IntegerDigits[#]], IntegerQ[(Sqrt[1+8#]-1)/2]&]]>0&]  (* Marco Ripà, Nov 07 2022 *)

A large number of errors corrected by N. J. A. Sloane, Apr 15 1996

Edited, corrected and extended by R. J. Mathar, Jan 08 2008

A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 175, 186, 198, 211, 225, 240, 240, 241, 243, 246, 250, 255, 261, 268, 276, 285, 295, 306, 318, 331, 345, 360, 360, 361, 363, 366, 370, 375, 381, 388

Offset: 0

Hieronymus Fischer, Jun 11 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A130486, A010872, A010873, A010874, A010875, A010876, A010878, A130481, A130482, A130483, A130484, A130485, A130487.

Accumulate[Mod[Range[0,60],16]] (* Harvey P. Dale, May 30 2020 *)

a(n)=120*floor(n/16)+A130909(n)*(A130909(n)+1)/2. - G.f.: g(x)=(sum{1<=k<16, k*x^k})/((1-x^16)(1-x)). Also: g(x)=x(15x^16-16x^15+1)/((1-x^16)(1-x)^3).

a(n) = +a(n-1) +a(n-16) -a(n-17). G.f. ( x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +14*x^13 +15*x^14) ) / ( (1+x) *(1+x^2) *(1+x^4) *(1+x^8) *(x-1)^2 ). - R. J. Mathar, Nov 05 2011

A134804 Remainder of triangular number A000217(n) modulo 9.

Original entry on oeis.org

0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6

Offset: 0

R. J. Mathar, Jan 28 2008

Periodic with period 9 since A000217(n+9) = A000217(n)+9(n+5) .

From Jacobsthal numbers A001045, A156060 = 0,1,1,3,5,2,3,7,4,0,8, = b(n). a(n)=A156060(n)*A156060(n+1) mod 9. Same transform (a(n)*a(n+1) mod 9 or b(n)*b(n+1) mod 9) in A157742, A158012, A158068, A158090. - Paul Curtz, Mar 25 2009

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 3, 6, 1, 6, 3, 1, 0},105] (* Ray Chandler, Aug 26 2015 *)

a(n) = A010878(A000217(n)) = A010878(A055263(n)) = a(n-9).

O.g.f.: (-2x+2)/[3(x^2+x+1)]+(-3+3x^5)/(x^6+x^3+1)-7/[3(x-1)].

A169821 a(n) = n*n in carryless arithmetic mod 9 in base 10.

Original entry on oeis.org

0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 100, 121, 144, 160, 187, 117, 130, 154, 171, 100, 400, 441, 484, 430, 477, 427, 460, 414, 451, 400, 0, 61, 34, 0, 67, 37, 0, 64, 31, 0, 700, 781, 774, 760, 757, 747, 730, 724, 711, 700, 700, 711, 724, 730, 747, 757, 760, 774, 781, 700, 0, 31, 64

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 11 2010

Addition and multiplication are the same as in school, that is, done in base 10, except that there are no carries and individual digits are added or multiplied mod 9.

			13*13 = 160:
..13
..13
----
..30
.13.
----
.160
----

Index entries for sequences related to carryless arithmetic

Cf. A010878, A169908. See A170990 for a better version.

A140816 A third of digital roots of Bernoulli number denominators.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1

Offset: 0

Paul Curtz, Jul 16 2008

a(n)= A010878(A140814(n))/ 3.

Edited and extended by R. J. Mathar, Jul 29 2008

A194597 Digital roots of the nonzero hexagonal numbers.

Original entry on oeis.org

1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3, 1, 3, 9, 1, 6, 6, 1, 9, 3

Offset: 1

Ant King, Aug 30 2011

This is a periodic sequence with period 9 and cycle 1,6,6,1,9,3,1,3,9 - which are also the coefficients of x in the numerator of the generating function.

a(n) = A010888(A000384(n)). - Reinhard Zumkeller, Jan 09 2013

			The sixth nonzero hexagonal number is A000384(6)=66. As 6+6=12 and 1+2=3, this has digital root 3 and so a(6)=3.

Cf. A000384.

Cf. A010878, A010888.

a194597 n = [1,6,6,1,9,3,1,3,9] !! a010878 (n-1)
-- Reinhard Zumkeller, Jan 09 2013

&cat[ [1,6,6,1,9,3,1,3,9]: k in [1..10] ]; // Vincenzo Librandi, Aug 11 2015

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[#]&,n];DigitalRoot[ # (2#-1)]&/@Range[63]
CoefficientList[Series[(1 + 6 x + 6 x^2 + x^3 + 9 x^4 + 3 x^5 + x^6 + 3 x^7 + 9 x^8)/((1 - x) (1 + x + x^2) (1 + x^3 + x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 11 2015 *)
PadRight[{},120,{1,6,6,1,9,3,1,3,9}] (* Harvey P. Dale, Oct 02 2018 *)

a(n) = a(n-9), and as the sum of the terms contained in each cycle is 39 they also satisfy the eighth-order inhomogeneous recurrence a(n) = 39 - a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - a(n-6) - a(n-7) - a(n-8).
a(n) = 2 + cos(2/9*(n-5)*Pi) + cos(4/9*(n-5)*Pi) + cos(2/3*(n-5)*Pi) + cos(8/9*(n-5)*Pi) + cos(4/3*(n-5)*Pi) + cos(14/9*(n-5)*Pi) + cos(16/9*(n-5)*Pi) + cos((2 n Pi)/9) + cos((4 n Pi)/9) + cos((2 n Pi)/3) + cos((8 n Pi)/9) + cos((10 n Pi)/9) + cos((4 n Pi)/3) + cos((14 n Pi)/9) + cos((16 n Pi)/9) + cos(2/9 (2+5 n) Pi) + (8n + 5n^2 + 7n^3 + n^5 + n^7 + 6n^8) mod 9.
G.f.: x(1+6x+6x^2+x^3+9x^4+3x^5+x^6+3x^7+9x^8)/((1-x)(1+x+x^2)(1+x^3+x^6)).

A240597 Numbers k such that sigma(k) == k (mod 9).

Original entry on oeis.org

1, 15, 24, 42, 60, 64, 69, 78, 90, 100, 114, 123, 133, 147, 153, 177, 186, 198, 222, 231, 240, 258, 259, 270, 276, 288, 289, 306, 339, 360, 366, 393, 402, 403, 414, 429, 438, 447, 459, 474, 477, 492, 495, 501, 507, 511, 522, 582, 588, 594, 600

Offset: 1

Ivan N. Ianakiev, Sep 13 2014

That is, numbers k that satisfy the following:
A010878(k) = A105852(k) or A010878(k) = A010878(A000203(k)).
A010888(k) = A190998(k) or A010888(k) = A010888(A000203(k)).

			sigma(15) = 24. 24 == 15 (mod 9), therefore 15 is in the sequence.

Ivan N. Ianakiev, Table of n, a(n) for n = 1..10000

Cf. A000203, A190998, A010878, A105852, A010888.

Select[Range[1000],Mod[#,9]==Mod[DivisorSigma[1,#],9]&]

A010888(a(n)) = A010888(A000203(a(n))).

A010888(a(n)) = A190998(a(n)).

A277547 a(n) = n/9^m mod 9, where 9^m is the greatest power of 9 that divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 2, 3, 4, 5, 6, 7, 8, 3, 1, 2, 3, 4, 5, 6, 7, 8, 4, 1, 2, 3, 4, 5, 6, 7, 8, 5, 1, 2, 3, 4, 5, 6, 7, 8, 6, 1, 2, 3, 4, 5, 6, 7, 8, 7, 1, 2, 3, 4, 5, 6, 7, 8, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5

Offset: 1

Clark Kimberling, Oct 19 2016

a(n) is the rightmost nonzero digit in the base 9 expansion of n.

			a(11) = (11/9 mod 9) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A010878.

Table[Mod[n/9^IntegerExponent[n, 9], 9], {n, 1, 160}]

a(n) = n/9^valuation(n, 9) % 9; \\ Michel Marcus, Oct 20 2016

cp's OEIS Frontend

A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

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A141726 Sawtooth with period length 9: repeat 8, 7, 6, 5, 4, 3, 2, 1, 0.

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A007960 Positive numbers k with the property that some permutation of the digits of k is a triangular number.

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A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

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A134804 Remainder of triangular number A000217(n) modulo 9.

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A169821 a(n) = n*n in carryless arithmetic mod 9 in base 10.

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A140816 A third of digital roots of Bernoulli number denominators.

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A194597 Digital roots of the nonzero hexagonal numbers.

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