cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A101493 Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).

Original entry on oeis.org

1, 6, 5, 15, 14, 9, 28, 27, 22, 13, 45, 44, 39, 30, 17, 66, 65, 60, 51, 38, 21, 91, 90, 85, 76, 63, 46, 25, 120, 119, 114, 105, 92, 75, 54, 29, 153, 152, 147, 138, 125, 108, 87, 62, 33, 190, 189, 184, 175, 162, 145, 124, 99, 70, 37, 231, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

Keywords

Comments

The triangle is generated from the product B*A of the infinite lower triangular matrices A =
1 0 0 0 ...
1 1 0 0 ...
1 1 1 0 ...
1 1 1 1 ...
... and B =
1 0 0 0 ...
1 5 0 0 ...
1 5 9 0 ...
1 5 9 13 ...
...
T(n+0,0) = n*(2*n-1) = A000384(n) (Hexagonal numbers)
since T(n,n) = 4*n+1 = A016813(n).
T(n,n) = 4*n + 1 = A016813(n);
T(n+1,n) = 8*n + 6 = A017137(n);
T(n+2,n) = 12*n + 3 = A017557(n);
T(n,n)*T(n,0) = (n+1)*(2*n+1)*(4*n+1) = A079588(n).

Examples

			Triangle begins:
   1;
   6,  5;
  15, 14,  9;
  28, 27, 22, 13;
  45, 44, 39, 30, 17;
  66, 65, 60, 51, 38, 21;

Links

Crossrefs

Row sums give 10-gonal pyramidal numbers: n(n+1)(8n-5)/6 = A007585(n+1).
Cf. A101492 (for product A*B), A007585, A000384.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->(n+1)*(2*n+1)-k*(2*k-1)))); # Muniru A Asiru, Mar 05 2019
  • PARI
    T(n,k)=if(k>n,0,(n+1)*(2*(n+1)-1)-k*(2*k-1))
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0

Original entry on oeis.org

1, 3, -1, 5, -6, 1, 7, -15, 11, -1, 9, -28, 38, -20, 1, 11, -45, 90, -90, 37, -1, 13, -66, 175, -260, 207, -70, 1, 15, -91, 301, -595, 707, -469, 135, -1, 17, -120, 476, -1176, 1862, -1848, 1052, -264, 1, 19, -153, 708, -2100, 4158, -5502, 4692, -2340, 521, -1
Offset: 0

Views

Author

Floor van Lamoen, Nov 04 2005

Keywords

Comments

The sequence a(m) is also the expansion of (1-x^n)/(1-x-2x^n+x^{n+1}).
Instead of b(i) = a(n*i) one can take b(i) = a(n*i+p) for p=1..n-1.

Examples

			For n=5 (A113444) the recurrence relation is b(i) = 11b(i-1)-45b(i-2) +90b(i-3)-90b(i-4)+37b(i-5)-b(i-6), so the fifth row reads 11, -45, 90, -90, 37, -1.

Links

Crossrefs

Columns k=0..2 give: A005408, -A000384, A007585(n-1) for n>=1. - Alois P. Heinz, Jul 16 2009

Programs

  • Maple
    T:= (n,k)-> (-1)^k /(k+1)! *(1+k +(n-k) *2^(k+1)) *mul (n+j-k, j=1..k):
seq(seq(T(n,k), k=0..n), n=0..11);  # Alois P. Heinz, Jul 16 2009

Formula

From Alois P. Heinz, Jul 16 2009: (Start)
T(n,k) = (-1)^k/(k+1)! * (1+k+(n-k)*2^(k+1)) * Product_{j=1..k}(n+j-k).
G.f. of column k: (-1)^k * x^k * (1+(2^(k+1)-1)*x)/(1-x)^(k+2). (End)

Extensions

More terms from Alois P. Heinz, Jul 16 2009

A218330 Odd decagonal pyramidal numbers.

Original entry on oeis.org

1, 11, 175, 301, 1005, 1375, 3003, 3745, 6681, 7923, 12551, 14421, 21125, 23751, 32915, 36425, 48433, 52955, 68191, 73853, 92701, 99631, 122475, 130801, 158025, 167875, 199863, 211365, 248501, 261783, 304451, 319641, 368225, 385451, 440335, 459725, 521293
Offset: 1

Views

Author

Ant King, Oct 29 2012

Keywords

Examples

			The sequence of decagonal pyramidal numbers A007585 begins 0, 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375,... As the third odd term is 175, then a(3) = 175.

Links

Crossrefs

Cf. A007585, A218331.

Programs

  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1}, {1,11,175,301,1005,1375,3003}, 37]

Formula

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 512.
a(n) = (16*n-4*(-1)^n-17)*(4*n-(-1)^n-3)*(4*n-(-1)^n-1)/24.
G. f. x*(1+10*x+161*x^2+96*x^3+215*x^4+22*x^5+7*x^6)/((1-x)^4*(1+x)^3).

A218331 Even, nonzero decagonal pyramidal numbers.

Original entry on oeis.org

38, 90, 476, 708, 1826, 2366, 4600, 5576, 9310, 10850, 16468, 18700, 26586, 29638, 40176, 44176, 57750, 62826, 79820, 86100, 106898, 114510, 139496, 148568, 178126, 188786, 223300, 235676, 275530, 289750, 335328, 351520, 403206, 421498, 479676, 500196
Offset: 1

Views

Author

Ant King, Oct 29 2012

Keywords

Examples

			The sequence of nonzero decagonal pyramidal numbers begins 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375,... As the third even term is 476, then a(3) = 476.

Links

Crossrefs

Cf. A007585, A218330.

Programs

  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1},{38,90,476,708,1826,2366,4600},36]

Formula

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 512.
a(n) = (16*n-4*(-1)^n-1)*(4*n-(-1)^n+3)*(4*n-(-1)^n+1)/24.
G. f. 2*x*(19+26*x+136*x^2+38*x^3+37*x^4)/((1-x)^4*(1+x)^3).

A269441 Alternating sum of 10-gonal (or decagonal) pyramidal numbers.

Original entry on oeis.org

0, -1, 10, -28, 62, -113, 188, -288, 420, -585, 790, -1036, 1330, -1673, 2072, -2528, 3048, -3633, 4290, -5020, 5830, -6721, 7700, -8768, 9932, -11193, 12558, -14028, 15610, -17305, 19120, -21056, 23120, -25313, 27642, -30108, 32718, -35473, 38380, -41440, 44660
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2016

Keywords

Links

Crossrefs

Cf. A000292, A002717, A007585, A173196, A266677, A269428, A269429.

Programs

  • Magma
    [((-1)^n*(16*n^3+30*n^2-4*n-9)+9)/24: n in [0..40]]; // Vincenzo Librandi, Feb 27 2016
  • Mathematica
    Table[((-1)^n (16 n^3 + 30 n^2 - 4 n - 9) + 9)/24, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 10, -28, 62}, 41]

Formula

G.f.: x*(1 - 7*x)/((x - 1)*(x + 1)^4).
a(n) = ((-1)^n*(16*n^3 + 30*n^2 - 4*n - 9) + 9) /24.
a(n) = Sum_{k = 0..n} (-1)^k*A007585(k).
Sum_{n>=1} 1/a(n) = -0.9251958836055717745244669... . - Vaclav Kotesovec, Feb 26 2016
Previous Showing 11-15 of 15 results.

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