A258128
Octagonal numbers (A000567) that are the sum of two consecutive octagonal numbers.
Original entry on oeis.org
5461, 813281, 7272157205, 1083057360705, 9684433559760981, 1442322650052752161, 12896895753596262561301, 1920761265591267733640321, 17174976631595008767000306005, 2557904668044167195987033355105, 22872156829955018609383449248248341
Offset: 1
5461 is in the sequence because Oct(43) = 5461 = 2640 + 2821 = Oct(30) + Oct(31).
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CoefficientList[Series[-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1,1331714,-1331714,-1,1},{5461,813281,7272157205,1083057360705,9684433559760981},20] (* Harvey P. Dale, Feb 19 2018 *)
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Vec(-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)) + O('x^20))
A258129
Octagonal numbers (A000567) that are the sum of three consecutive octagonal numbers.
Original entry on oeis.org
698901, 5102520783381, 37252493940331837461, 271973082264557457061125141, 1985621622943208359132836202790421, 14496630316026749501691464257547633057301, 105837027604506739193825102426073141683789429781, 772695182809023513889440668692977953487035688873891861
Offset: 1
698901 is in the sequence because Oct(483) = 698901 = 231296 + 232965 + 234640 = Oct(278) + Oct(279) + Oct(280).
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CoefficientList[Series[-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{7300803,-7300803,1},{698901,5102520783381,37252493940331837461},20] (* Harvey P. Dale, Sep 16 2018 *)
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Vec(-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)) + O('x^20))
A258130
Octagonal numbers (A000567) that are the sum of ten consecutive octagonal numbers.
Original entry on oeis.org
1045, 1325345, 1910970885, 2755618515265, 3973599987865685, 5729928426883626945, 8262552817966202013445, 11914595433578836419585185, 17180838352667864150839647765, 24774756989951626526674352316385, 35725182398671892783600265200403845
Offset: 1
1045 is in the sequence because Oct(19) = 1045 = 1+8+21+40+65+96+133+176+225+280 = Oct(1) + ... + Oct(10).
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CoefficientList[Series[-95 x (63 x^2 - 1922 x + 11)/((x - 1) (x^2 - 1442 x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1443,-1443,1},{1045,1325345,1910970885},20] (* Harvey P. Dale, Jul 25 2019 *)
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Vec(-95*x*(63*x^2-1922*x+11)/((x-1)*(x^2-1442*x+1)) + O('x^20))
A258132
Octagonal numbers (A000567) that are the sum of fifteen consecutive octagonal numbers.
Original entry on oeis.org
4715040, 8463840, 1122749669280, 2015496399840, 267373851637578960, 479974343542849680, 63672943775553479639280, 114302050117965712710960, 15163202176330482896520455040, 27220118818712616412771202880, 3610995292612020914167620625112640
Offset: 1
4715040 is in the sequence because Oct(1254) = 4715040 = 300833 + 302736 + 304645 + 306560 + 308481 + 310408 + 312341 + 314280 + 316225 + 318176 + 320133 + 322096 + 324065 + 326040 + 328021 = Oct(317) + ... + Oct(331).
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CoefficientList[Series[-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
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Vec(-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)) + O('x^20))
Showing 1-4 of 4 results.