A122793 -id:A122793 - cp's OEIS frontend

A122797 A P_3-stuttered arithmetic progression with a(n+1) = a(n) if n is a triangular number, a(n+1) = a(n) + 1 otherwise.

1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_3(i) = the i-th triangular number.
As a triangle [1; 1,2; 2,3,4; ...], row sums = A064808: (1, 3, 9, 22, 45, 81, ...). - Gary W. Adamson, Aug 10 2007
a(n) = n - A003056(n-1). - Reinhard Zumkeller, Feb 12 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]

Cf. A001614, A122793, A122794, A122795, A122796, A122798, A122799, A122800.

Cf. A064808.

a122797 n = a122797_list !! (n-1)
a122797_list  = 1 : zipWith (+) a122797_list (map ((1 -) . a010054) [1..])
-- Reinhard Zumkeller, Feb 12 2012

nxt[{n_,a_}]:={n+1,If[OddQ[Sqrt[8n+1]],a,a+1]}; NestList[nxt,{1,1},100][[All,2]] (* Harvey P. Dale, Oct 10 2018 *)

isTriang(n) = {if (! issquare(8*n+1), return (0)); return (1);}
lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! isTriang(i), aa = aa + 1););} \\ Michel Marcus, Apr 01 2013

from math import isqrt
def A122797(n): return n+1-(k:=isqrt(m:=n<<1))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = A001614(n) - n + 1.

Definition corrected by Michel Marcus, Apr 01 2013

A122798 A P_5-stuttered arithmetic progression with a(n+1) = a(n) if n is a pentagonal number, a(n+1) = a(n)+4 otherwise.

Original entry on oeis.org

1, 1, 5, 9, 13, 13, 17, 21, 25, 29, 33, 37, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 181, 185, 189, 193, 197, 201, 205, 209

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_5(i) = the i-th pentagonal number.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122799, A122800.

nxt[{n_,a_}]:={n+1,If[IntegerQ[(1+Sqrt[24n+25])/6],a,a+4]}; Join[{1}, Transpose[ NestList[nxt,{1,1},60]][[2]]] (* Harvey P. Dale, May 07 2015 *)
nxt[{n_,a_}]:=With[{pn=PolygonalNumber[5,Range[0,30]]},{n+1,If[MemberQ[pn,n],a,a+4]}]; NestList[nxt,{1,1},100][[;;,2]] (* Harvey P. Dale, Sep 28 2023 *)

lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! ispolygonal(i, 5), aa += 4););} \\ Michel Marcus, Apr 01 2013, May 02 2015

a(n) = A045929(n) - n + 1.

Definition corrected by Michel Marcus, Apr 01 2013

A122800 A P_4-stuttered arithmetic progression with a(n+1)=a(n) if n is square, a(n+1)=a(n)+2 otherwise.

Original entry on oeis.org

1, 1, 3, 5, 5, 7, 9, 11, 13, 13, 15, 17, 19, 21, 23, 25, 25, 27, 29, 31, 33, 35, 37, 39, 41, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_4(i) = the i-th square.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122798, A122799.

nxt[{n_,a_}]:={n+1,If[IntegerQ[Sqrt[n+1]],a,a+2]}; NestList[nxt,{0,1},100][[All,2]] (* Harvey P. Dale, Jan 01 2020 *)

lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! issquare(i), aa = aa + 2););} \\ Michel Marcus, Apr 01 2013

a(n) = A045928(n)-n+1.

A122794 Connell (3,2)-sum sequence (partial sums of the (3,2)-Connell sequence).

Original entry on oeis.org

1, 3, 8, 16, 25, 37, 52, 70, 91, 113, 138, 166, 197, 231, 268, 308, 349, 393, 440, 490, 543, 599, 658, 720, 785, 851, 920, 992, 1067, 1145, 1226, 1310, 1397, 1487, 1580, 1676, 1773, 1873, 1976, 2082, 2191, 2303, 2418, 2536, 2657, 2781, 2908, 3038, 3171, 3305, 3442, 3582, 3725, 3871, 4020, 4172, 4327, 4485, 4646, 4810, 4977, 5147, 5320, 5496, 5673, 5853, 6036, 6222, 6411, 6603

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045928, A001614, A045929, A045930.

Cf. A122793, A122795, A122796, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122800).

A122795 Connell (5,3)-sum sequence (partial sums of the (5,3)-Connell sequence).

Original entry on oeis.org

1, 3, 10, 22, 39, 57, 80, 108, 141, 179, 222, 270, 319, 373, 432, 496, 565, 639, 718, 802, 891, 985, 1080, 1180, 1285, 1395, 1510, 1630, 1755, 1885, 2020, 2160, 2305, 2455, 2610, 2766, 2927, 3093, 3264, 3440, 3621, 3807, 3998, 4194, 4395, 4601, 4812, 5028, 5249, 5475, 5706, 5938, 6175, 6417, 6664, 6916, 7173, 7435, 7702, 7974, 8251, 8533, 8820, 9112, 9409, 9711, 10018, 10330, 10647, 10969

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045929, A001614, A045928, A045930.

Cf. A122793, A122794, A122796, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122798).

A122796 Connell (3,5)-sum sequence (partial sums of the (3,5)-Connell sequence).

Original entry on oeis.org

1, 3, 8, 16, 27, 41, 58, 76, 97, 121, 148, 178, 211, 247, 286, 328, 373, 421, 470, 522, 577, 635, 696, 760, 827, 897, 970, 1046, 1125, 1207, 1292, 1380, 1471, 1565, 1660, 1758, 1859, 1963, 2070, 2180, 2293, 2409, 2528, 2650, 2775, 2903, 3034, 3168, 3305, 3445, 3588, 3734, 3883, 4035, 4190, 4346, 4505, 4667, 4832, 5000, 5171, 5345, 5522, 5702, 5885, 6071, 6260, 6452, 6647, 6845

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.

Cf. A045930, A001614, A045928, A045929.

Cf. A122793, A122794, A122795, A122797, A122798, A122799, A122800.

a(n) = (n-th triangular number)-n+(n-th partial sum of A122799).

A122799 A P_7-stuttered arithmetic progression with a(n+1)=a(n) if n is not a heptagonal number, a(n+1)=a(n)+2 otherwise.

Original entry on oeis.org

1, 1, 3, 5, 7, 9, 11, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187

Offset: 1

Grady Bullington (bullingt(AT)uwosh.edu), Sep 14 2006

P_7(i) = the i-th heptagonal number.

Douglas E. Iannucci and Donna Mills-Taylor. On Generalizing the Connell Sequence. Journal of Integer Sequences 2 (1999), Article 99.1.7.
Grady D. Bullington, The Connell Sum Sequence, J. Integer Seq. 10 (2007), Article 07.2.6. (includes direct formula for a(n))

Cf. A001614, A122793, A122794, A122795, A122796, A122797, A122798, A122800.

isHeptag(n) = {if (! issquare(40*n+9, &res), return (0)); if ((res + 3) % 10, return (0), return (1));}
lista(m) = {aa = 1; for (i=1, m, print1(aa, ", "); if (! isHeptag(i), aa += 2););} \\ Michel Marcus, Apr 01 2013

a(n) = A045930(n)-n+1.

Definition corrected by Michel Marcus, Apr 01 2013

A337300 Partial sums of the geometric Connell sequence A049039.

Original entry on oeis.org

1, 3, 7, 12, 19, 28, 39, 51, 65, 81, 99, 119, 141, 165, 191, 218, 247, 278, 311, 346, 383, 422, 463, 506, 551, 598, 647, 698, 751, 806, 863, 921, 981, 1043, 1107, 1173, 1241, 1311, 1383, 1457, 1533, 1611, 1691, 1773, 1857, 1943, 2031, 2121, 2213, 2307, 2403

Offset: 1

Kevin Ryde, Aug 22 2020

a(n) is Newey's "more complicated" conjectured length of the shortest sequence containing all permutations of 1..n (A062714). It agrees with A062714(n) for n <= 7. [But not for n=8. - Pontus von Brömssen, Aug 18 2025]

Kevin Ryde, Table of n, a(n) for n = 1..5000
Malcolm Newey, Notes On a Problem Involving Permutations As Subsequences, Stanford Artificial Intelligence Laboratory, Memo AIM-190, STAN-CS-73-340, 1973. Conjectured M(n) formula bottom of page 12.

Cf. A000295, A049039 (first differences), A062714, A122793 (arithmetic Connell sums).

a(n) = my(k=logint(n,2)); n^2 - k*(n+1) + (2<

a(n) = n^2 - k*n + F(k) where k = floor(log_2(n)) and F(0) = 0 then F(k) = k + 2*F(k-1) [Newey], which is F(k) = 2^(k+1) - k - 2 = A000295(k+1), the Eulerian numbers.
a(n) = n^2 - k*(n+1) + 2*(2^k - 1) where k = floor(log_2(n)).
G.f.: 2*x/(1-x)^3 - ( Sum_{j>=0} x^(2^j) )/(1-x)^2.
a(n) = Sum_{i=1..n} A049039(i). - Gerald Hillier, Jun 18 2016

cp's OEIS Frontend

A122797 A P_3-stuttered arithmetic progression with a(n+1) = a(n) if n is a triangular number, a(n+1) = a(n) + 1 otherwise.

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A122798 A P_5-stuttered arithmetic progression with a(n+1) = a(n) if n is a pentagonal number, a(n+1) = a(n)+4 otherwise.

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A122800 A P_4-stuttered arithmetic progression with a(n+1)=a(n) if n is square, a(n+1)=a(n)+2 otherwise.

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A122794 Connell (3,2)-sum sequence (partial sums of the (3,2)-Connell sequence).

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A122795 Connell (5,3)-sum sequence (partial sums of the (5,3)-Connell sequence).

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A122796 Connell (3,5)-sum sequence (partial sums of the (3,5)-Connell sequence).

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A122799 A P_7-stuttered arithmetic progression with a(n+1)=a(n) if n is not a heptagonal number, a(n+1)=a(n)+2 otherwise.

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A337300 Partial sums of the geometric Connell sequence A049039.

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Formula